Theory for Moisture Transport in Porous Media
In a porous material, moisture may be present under vapor and liquid state, and moisture transport occurs through convection and diffusion, liquid water being fixed on the pores walls mainly in two ways:
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Adsorption: the vapor is fixed under molecular films on the pores walls. Equilibrium vapor pressure is reached in the pores under thermodynamic conditions that are significantly different than those necessary in a free medium. Thus saturation conditions are changed, and the relative humidity is significantly changed as well, becoming a relevant variable for the characterization of moisture content.
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Capillarity: when the porous matrix is wetted by liquid water, there is a pressure difference between the liquid and gas (moist air) phases, due to the curvature of the wetting interfaces in each pore. The pressure difference, called capillary pressure, becomes the driving potential for a further filling of the pores by the water. Note that in some applications, the capillary pressure is replaced by the suction.
The predominance of adsorption or capillarity depends both on the amount of liquid water present in the porous medium, and on the size and shape of the pores, and should be defined in function of the material and process under study. Two moisture regions can be distinguished:
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Hygroscopic region: adsorption is the predominant phenomenon, and the equilibrium moisture content is better characterized from the relative humidity, through the specification of a sorption curve. For relative humidity values below 0.95, the slope of the sorption curve is relatively small, and the moisture content increases in a nearly linear way with relative humidity. Building materials are most of the time in this moisture region. Above 0.95, a very small variation of relative humidity induces a large variation of the moisture content, and the capillary pressure is best suited to characterize the moisture content.
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Capillary water region: capillarity is the predominant phenomenon, and the equilibrium moisture content is better characterized from the capillary pressure, through the specification of a retention curve (the van Genuchten and Brooks & Corey retention models for example).
The relative humidity and capillary pressure are called state variables: they describe the thermodynamic state of water in the pore space under two phases in equilibrium. At equilibrium, they are related through Kelvin’s law, which is a relation of thermodynamical origin. As long as equilibrium is satisfied, the state variables are continuous between two media (at the interface between two porous media with different porosities, or between a porous and a free medium), and Kelvin’s law is valid on both sides of the interface. In opposition, the liquid saturation or the moisture content depend on the porous matrix.
Note that capillary transport of liquid water should be considered in both regions.
The Moisture Transport in Porous Media Interface is dedicated to the modeling of porous media in the hygroscopic moisture region. The total moisture content is thus defined by a sorption curve as
where the liquid saturation, sl, describes the amount of liquid water within the pores, and εp (dimensionless) is the porosity.
By summing the mass conservation equations for liquid water and vapor, and equilibrium between the liquid and gas phases, a single equation for the transport of the total moisture can be written, in which the evaporation and condensation source terms of each equation cancel out:
(4-137)
with the following material properties, fields, and source:
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w(ϕw) (SI unit: kg/m3) is the total moisture content.
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  ϕw (dimensionless) is the relative humidity.
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 ρg (SI unit: kg/m3) is the moist air density, defined from the dry air and vapor densities, in function of the amount of vapor.
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 ug (SI unit: m/s) is the moist air velocity field, that should be interpreted as the Darcy velocity, that is, the volume flow rate per unit cross sectional area.
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 ωv (dimensionless) is the vapor mass fraction in moist air, defined as
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 Μv (SI unit: kg/mol) is the molar mass of water vapor.
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 csat (SI unit: mol/m3) is the vapor saturation concentration.
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 T (SI unit: K) is the temperature.
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 gw is the moisture diffusive flux is defined as:
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 Deff (SI unit: m2/s) is the effective vapor diffusion coefficient in the porous medium, computed from the diffusion coefficient in a free medium, and accounting for the porosity and tortuosity of the porous medium.
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 ul (SI unit: m/s) is the liquid water velocity field, defined from the absolute pressure gradient by the Darcy’s law as:
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 κrl (dimensionless) is the relative liquid water permeability, that may be defined as a function of the liquid saturation.
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 κ (SI unit: m2) is the porous medium permeability.
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 pA (SI unit: Pa) is the absolute pressure.
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 ρl (SI unit: kg/m3) is the liquid water density, defined as a function of the temperature.
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 glc is the liquid water capillary flux, defined either from the capillary pressure gradient by a Darcy’s law:
or by a diffusion equation using the relative humidity:
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 Dw (SI unit: m2/s) is the moisture diffusivity.
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 pc is the capillary pressure, that can be related to the relative humidity using Kelvin’s law:
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 G (SI unit: kg/(m3⋅s)) is a moisture source (or sink). See the Moisture Source node.
The moisture source due to evaporation may be obtained from the transport equation for vapor:
This moisture source is added as a mass source in the fluid flow equations solved for the computation of the moist air velocity field, ug.
The corresponding latent heat source is defined as:
where Lv is the latent heat of evaporation.
For a steady-state problem, the relative humidity does not change with time and the first term in the moisture transport equation disappears.